CONGRUENT TRIANGLES
6.1.1 – 6.1.4
Two triangles are congruent if there is a sequence of rigid transformations that carry one onto the other. Two triangles are also congruent if they are similar figures with a ratio of similarity of 1, that is 11. One way to prove triangles congruent is to prove they are similar first, and then prove that the ratio of similarity is 1. In these lessons, students find short cuts that enable them to prove triangles congruent in fewer steps, by developing five triangle congruence conjectures. They are SSS ≅, ASA ≅, AAS ≅, SAS ≅, and HL ≅, illustrated below.
Note: “S” stands for “side” and “A” stands for “angle.” HL ≅ is only used with right triangles. The “H” stands for “hypotenuse” and the “L” stands for leg. The pattern appears to be “SSA” but this arrangement is NOT one of our conjectures, since it is only true for right triangles.
See the Math Notes boxes in Lessons 6.1.1 and 6.1.4. SSS ≅
ASA ≅
AAS ≅ SAS ≅
Example 1
Use your triangle congruence conjectures to decide whether or not each pair of triangles must be congruent. Base each decision on the markings, not on appearances. Justify each answer.
a. b.
c. d.
e. f.
In part (a), the triangles are congruent by the SAS ≅ conjecture. The triangles are also congruent in part (b), this time by the SSS ≅ conjecture. In part (c), the triangles are congruent by the AAS ≅ conjecture. Part (d) shows a pair of triangles that are not necessarily congruent. The first triangle displays an ASA arrangement, while the second triangle displays an AAS arrangement. The triangles could still be congruent, but based on the markings, we cannot conclude that they definitely are congruent. The triangles in part (e) are right triangles and the markings fit the HL ≅ conjecture. Lastly, in part (f), the triangles are congruent by the ASA ≅ conjecture.
Example 2
Using the information given in the diagrams below, decide if any triangles are congruent, similar but not congruent, or not similar. If you claim the triangles are congruent or similar, create a flow chart justifying your answer.
a. b.
In part (a), ∆ABD ≅ ∆CBD by the SAS ≅ conjecture. Note: If you only see “SA,” observe that BD is congruent to itself. The Reflexive Property justifies stating that something is equal or congruent to itself.
In part (b), ∆WXV ~ ∆ZYV by the AA ~ conjecture. The triangles are not necessarily congruent; they could be congruent, but since we only have information about angles, we cannot conclude anything else. W X Y Z V ∆WXV ~ ∆ZYV Vertical angles = AA ~ A B C D AD = CD Given m∠ADB = m∠CDB Given BD = BD ∆ABD ≅∆CBD Reflexive Prop. SAS ≅
Problems
Briefly explain if each of the following pairs of triangles are congruent or not. If so, state the triangle congruence conjecture that supports your conclusion.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
Use your triangle congruence conjectures to decide whether or not each pair of triangles must be congruent. Base your decision on the markings, not on appearances. Justify your answer.
16. A 17. B C D E P K M L N P Q O S R U X T W V X Y W Z G J H I C B A E D F V Y X U T W Q R S T A C D B F E G I L H J K P M N O 12 A B D C 13 5 H E G I 5 4 3 R P Q J K L 39º 112º 112º 39º 8 F A B C D E 12 4 12 8 4
18. 19.
20. 21.
Using the information given in each diagram below, decide if any triangles are congruent, similar but not congruent, or not similar. If you claim the triangles are congruent or similar, create a flowchart justifying your answer.
22. 23. 24. 25. V W X Y Z T E S A K L E B A R O N D A V I S L U N C H T K R I J A P E S
In each diagram below, are any triangles congruent? If so, prove it. (Note: Justify some using flowcharts and some by writing two-column proofs.)
26. 27. 28.
29. 30. 31.
Complete a proof for each problem below in the style of your choice.
32. Given: TR and MN bisect each other. 33. Given: CD bisects ∠ACB; ∠1 ≅∠2.
Prove: ∆NTP≅∆MRP Prove: ∆CDA≅∆CDB
34. Given: ABCD, ∠B≅∠D, AB≅CD 35. Given: PG≅SG, TP≅TS
Prove: ∆ABF≅∆CDE Prove: ∆TPG≅∆TSG
36. Given: OE⊥MP, OE bisects ∠MOP 37. Given: ADBC, DCBA
Prove: ∆MOE≅∆POE Prove: ∆ADB≅∆CBD
B A D C B A C D E B A D C B A D C B A D C B A D C E F B A D C E F B A D C 1 2 B A D C N T M R P E M O G S P T
38. Given: AC bisects DE, ∠A≅∠C 39. Given: PQ⊥RS, ∠R≅∠S
Prove: ∆ADB≅∆CEB Prove: ∆PQR≅∆PQS
40. Given: ∠S≅∠R, PQ bisects ∠SQR 41. Given: TU≅GY, KY HU, KT ⊥TG,
Prove: ∆SPQ≅∆RPQ HG⊥TG. Prove: ∠K≅∠H
42. Given: MQWL, MQ≅WL
Prove: MLWQ
Consider the diagram at right. 43. Is ∆BCD≅∆EDC? Prove it! 44. Is AB≅DC? Prove it! 45. Is AB≅ED? Prove it! B C E D A Q P S R Q L M W P S Q R A C E D B K T Y U G H
Answers
1. ∆ABC≅∆DEF by ASA. 2. ∆GIH≅∆LJK by SAS.
3. ∆PNM≅∆PNO by SSS. 4. QS≅QS, so ∆QRS≅∆QTS by HL.
5. The triangles are not necessarily congruent.
6. ∆ABC≅∆DFE by ASA or AAS. 7. GI≅GI, so ∆GHI≅∆IJG by SSS. 8. Alternate interior angles = used twice, so ∆KLN≅∆NMK by ASA.
9. Vertical angles ≅ at 0, so ∆POQ≅∆ROS by SAS.
10. Vertical angles and/or alternate interior angles =, so ∆TUX≅∆VWX by ASA. 11. No, the length of each hypotenuse is different.
12. Pythagorean Theorem, so ∆EGH≅∆IHG by SSS.
13. Sum of angles of triangle = 180º, but since the equal angles do not correspond, the triangles are not congruent.
14. AF + FC = FC + CD, so ∆ABC≅∆DEF by SSS. 15. XZ≅XZ, so ∆WXZ≅∆YXZ by AAS.
16. ∆ABC≅∆EDC by AAS ≅
17. ∆PQS≅∆PRS by AAS ≅, with PS≅PS by the Reflexive Property.
18. ∆VXW≅∆ZXY by ASA ≅, with ∠VXW ≅ ∠ZXY because vertical angles are ≅. 19. ∆TEA≅∆SAE by SSS ≅, with EA≅EAby the Reflexive Property.
20. ∆KLB≅∆EBL by HL ≅, with BL≅BL by the Reflexive Property. 21. Not necessarily congruent.
22. ∆DAV ~ ∆ISV by SAS ~
23. ∆LUN and ∆HTC are not necessarily similar based on the markings. ∆DAV ~ ∆ISV Equal
markings Vert. angles =
24. ∆SAP ~ ∆SJE by AA~
25. ∆KRS≅∆ISR by HL ≅
26. Yes 27. Yes
28. Yes 29. Yes
30. Not necessarily. 31. Yes
AP || JE ∆SAP ~ ∆SJE given Alt. int. angles = Alt. int. angles = AA~
∆KRS & ∆ISR are right triangles. Given KR = IS Given RS = RS ∆KRS ≅ ∆ISR HL ≅ Refl. Prop. ∠BAD ≅ ∠BCD ∠BDC ≅ ∠BDA ΔABD ≅ ΔCBD Right ∠s are ≅ Given Reflexive Prop. AAS ≅
∠B ≅ ∠E ∠BCA ≅ ∠ECD
ΔABD ≅ ΔDEC Vertical ∠s are ≅ Given Given ASA ≅ ∠BDC ≅ ∠BDA ΔABC ≅ ΔDBC Right ∠s are ≅ Given Reflexive Prop. SAS ≅ ΔABC ≅ ΔCDA Given Given Reflexive Prop. SSS ≅
32. NP≅MP and TP≅RP by definition of bisector. ∠NPT≅∠MPR because vertical angles
are equal. So, ∆NTP≅∆MRP by SAS ≅.
33. ∠ACD≅∠BCD by definition of angle bisector. CD≅CD by reflexive so ∆CDA≅∆CDB
by ASA ≅.
34. ∠A≅∠C since alternate interior angles of parallel lines congruent so ∆ABF≅∆CDE by ASA ≅.
35. TG≅TG by reflexive so ∆TPG≅∆TSG by SSS ≅.
36. ∠MEO≅∠PEO because perpendicular lines form ≅ right angles ∠MOE≅∠POE by angle bisector and OE≅OE by reflexive. So, ∆MOE≅∆POE by ASA ≅.
37. ∠CDB≅∠ABD and ∠ADB≅∠CBD since parallel lines give congruent alternate interior angles. DB≅DB by reflexive so ∆ADB≅∆CBD by ASA ≅.
38. DB≅EB by definition of bisector. ∠DBA≅∠EBC since vertical angles are congruent. So ∆ADB≅∆CEB by AAS ≅.
39. ∠RQP≅∠SQP since perpendicular lines form congruent right angles. PQ≅PQ
(Reflexive Prop.) so ∆PQR≅∆PQS by AAS ≅.
40. ∠SQP≅∠RQP by angle bisector and PQ≅PQ by reflexive, so ∆SPQ≅∆RPQ by AAS ≅.
41. ∠KYT≅∠HUG because parallel lines form congruent alternate exterior angles.
TY + YU = YU + GU so TY ≅ GU by subtraction. ∠T≅∠G since perpendicular lines form congruent right angles. So ∆KTY≅∆HGU by ASA≅. Therefore, ∠K≅∠H since ≅
triangles have congruent parts.
42. ∠MQL≅∠WLQ since parallel lines form congruent alternate interior angles. QL≅QL by the Reflexive Property so ∆MQL≅∆WLQ by SAS ≅, then ∠WQL≅∠MLQ since
congruent triangles have congruent parts. So MLWQ since congruent alternate interior angles are formed by parallel lines.
43. Yes 44. Not necessarily. 45. Not necessarily. ∠BDC ≅ ∠ECD ΔBCD ≅ ΔEDC
Alt. Int. ∠s are ≅ Given
Reflexive Prop.